where the symmetric bilinear form \(a(w_h,v_h)=\int _\Omega \nabla w_h \cdot \nabla v_h \, \text{d}V\)  is continuous and coercive on \(H^1_0(\Omega )\) , and the linear form \(l(v_h)=\int _{\partial \Omega _N} g_N v_h \, \text{d}S-\int _\Omega \nabla \ell (g_D) \cdot \nabla v_h \, \text{d}V\)  is also continuous on \(H^1_0(\Omega )\) . These conditions are sufficient to guarantee well-posedness of the problem (REF ) to find \(w_h \in H^1_0(\Omega )\)  and \(u_h \in H^1(\Omega )\)  for sufficiently smooth boundary data [1]}, [2]}. Because of the coercivity of the bilinear form, it induces the operator norm \(\Vert  v_h \Vert ^2_{a} = a(v_h, v_h)\) , which is identical to the \(H^1_0(\Omega )\) -seminorm for the boundary-fitted finite element formulation of the Poisson problem.
